Journal of Commutative Algebra

Cofiniteness and non-vanishing of local cohomology modules

Iraj Bagheriyeh, Kamal Bahmanpour, and Jafar A'Zami

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $R$ be a commutative Noetherian local ring, $I$ an ideal of $R$, and let $M$ be a non-zero finitely generated $R$-module. In this paper, we establish some new properties of the local cohomology modules $H^i_I(M)$, $i\geq 0$. In particular, we show that if $(R,\mathfrak{m})$ is a Noetherian local integral domain of dimension $d \leq 4$ which is a homomorphic image of a Cohen-Macaulay ring and $x_1,\ldots,x_n$ is a part of a system of parameters for $R$, then for all $i\geq0$, the $R$-modules $H^i_{I}(R)$ are $I$-cofinite, where $I=(x_1,\ldots,x_n)$. Also, we prove that if $(R,\mathfrak{m})$ is a Noetherian local ring of dimension~$d$ and $x_1,\ldots,x_t$ is a part of a system of parameters for $R$, then $H^{d-t}_{\mathfrak{m}}(H^t_{(x_1,\ldots,x_t)}(R))\neq 0$. In particular, $\mu^{d-t}(\mathfrak{m},H^t_{(x_1,\ldots,x_t)}(R))\neq 0$ and ${\rm injdim}_R(H^t_{(x_1,\ldots,x_t)}(R))\geq d-t$.

Article information

Source
J. Commut. Algebra Volume 6, Number 3 (2014), 305-321.

Dates
First available in Project Euclid: 17 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.jca/1416233320

Digital Object Identifier
doi:10.1216/JCA-2014-6-3-305

Mathematical Reviews number (MathSciNet)
MR3278806

Zentralblatt MATH identifier
1299.13019

Subjects
Primary: 13D45: Local cohomology [See also 14B15] 14B15: Local cohomology [See also 13D45, 32C36]
Secondary: 13E05: Noetherian rings and modules

Keywords
Asso ciated primes cofinite mo dules Krull dimension lo cal cohomology

Citation

Bagheriyeh, Iraj; Bahmanpour, Kamal; A'Zami, Jafar. Cofiniteness and non-vanishing of local cohomology modules. J. Commut. Algebra 6 (2014), no. 3, 305--321. doi:10.1216/JCA-2014-6-3-305. https://projecteuclid.org/euclid.jca/1416233320.


Export citation

References

  • R. Abazari and K. Bahmanpour, Cofiniteness of extension functors of cofinite modules, J. Algebra 330 (2011), 507–516.
  • I. Bagheriyeh, J. A'zami and K. Bahmanpour, Generalization of the Lichtenbaum-Hartshorne vanishing theorem, Comm. Algebra 40 (2012), 134–137.
  • K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra 321 (2009), 1997–2011.
  • M.P. Brodmann and R.Y. Sharp, Local cohomology; An algebraic introduction with geometric applications, Cambridge University Press, Cambridge, 1998.
  • D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Alg. 121 (1997), 45–52.
  • A. Grothendieck, Local cohomology, Lect. Notes Math. 862, Springer, New York, 1966.
  • R. Hartshorne, Affine duality and cofiniteness, Inv. Math. 9 (1970), 145–164.
  • C. Huneke, Problems on local cohomology, Free resolutions in commutative algebra and algebraic geometry, Res. Notes Math. 2 (1992), 93–108.
  • K.-I. Kawasaki, On the finiteness of Bass numbers of local cohomology modules, Proc. Amer. Math. Soc. 124 (1996), 3275–3279.
  • R. Lü and Z. Tang, The $f$-depth of an ideal on a module, Proc. Amer. Math. Soc. 130 (2001), 1905–1912.
  • T. Marley, Finitely graded local cohomology and depths of graded algebra, Proc. Amer. Math. Soc. 123 (1995), 3601–3607.
  • ––––, The associated primes of local cohomology modules over rings of small dimension, Manuscr. Math. 104 (2001), 519–525.
  • T. Marley and J.C. Vassilev, Cofiniteness and associated primes of local cohomology modules, J. Algebra 256 (2002), 180–193.
  • H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, UK, 1986.
  • L. Melkersson, On asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Cambr. Phil. Soc. 107 (1990), 267–271.
  • L. Melkersson, Properties of cofinite modules and application to local cohomology, Math. Proc. Cambr. Phil. Soc. 125 (1999), 417–423.
  • T. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), 649–668.
  • P. Schenzel, Proregular sequences, local cohomology, and completion, Math. Scand. 92 (2003), 161–180.
  • K.I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997), 179–191.